Calculus for 5th Graders

“Calculus is the study of change” is a common description and all too common misunderstanding. For if calculus is the study of change then algebra becomes the “study of symbols?” I believe this confusion between algebra and calculus among math educators makes these subjects, and the choice to study them, a challenge to students. It is the reason some even advocate changing the traditional arithmetic-algebra-calculus sequence into arithmetic-algebra-statistics, believing statistics to be more important and more relevant to the everyday lives of students. I think they are wrong.

If we say instead: “Algebra is the study of change.” thenCalculus is the study of the rate of change.”

The symbol we associate with algebra, “x”, was defined by Descartes in 1637 as the symbol for a variable. Unfortunately, because today x also used to represent an unknown quantity as well as a variable, our students are confused about its meaning and the nature of algebra. For those who understand the language and concepts of mathematics, numbers represent fixed quantities, and variables like x represent changing quantities. Our students who ask, “What is x?” will likely describe math as abstract and early on make the decision that they are a member of the majority who just don’t get math.

Spreadsheets, work with discrete variables. A number or an unknown is a fixed quantity in a single cell. A variable is a row or column of numbers, a table of values. This fundamentally visual and concrete representation of variable makes the difference between an unknown and a variable vanish, and this all too often asked question, go away. Function, symbolized as f(x), often baffles students. A function can be visualized by the second column in a simple table, next to the first one, an “output” generated by a rule that transforms a variable (input) column into an output column.

We can graph the function easily on a spreadsheet. And we can make a table for its parameters so that we can change and adjust the function easily. This format gives students the ability to play with functions and to learn to visualize them and build models using them. For a mathematical model is built out of functions.

The derivative, a special class of functions, rates, ∆f(x)/∆x, could be a third column taking the differences between the function values and dividing by the variable differences. If a student makes ∆x, the interval between the input values, smaller and smaller she will be zooming in on the function and build an intuitive sense of limit and derivative. The derivative is now clearly not the study of change, it is the study of rate of change.

If rate is a concept we expect 5th and 6th graders to understand, then derivatives using discrete variables on spreadsheets is a concept they can understand as well. And they can be asked to solve real problems that ask for rates of change.

Spreadsheets enable all our students to get math, to see variables, functions, and derivatives as concrete easily calculated mathematics. And all can be easily envisioned graphically.

Try it yourself. Build a spreadsheet like this one and experiment by finding the slope of the curve at different points on the curve. Was Jim Kaput right. Can these fundamental and important ideas of algebra and even calculus be learned by 5th graders.